Large-Scale Oceanic Currents As Shallow-Water Asymptotic Solutions of the Navier-Stokes Equation in Rotating Spherical Coordinates ⁎ A
Deep-Sea Research Part II xxx (xxxx) xxx–xxx Contents lists available at ScienceDirect Deep-Sea Research Part II journal homepage: www.elsevier.com/locate/dsr2 Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates ⁎ A. Constantina, , R.S. Johnsonb a Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria b School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom ARTICLE INFO ABSTRACT Keywords: We show that a consistent shallow-water approximation of the incompressible Navier-Stokes equation written in Incompressible Navier-Stokes equation a spherical, rotating coordinate system produces, at leading order in a suitable limiting process, a general linear Ekman-type solution theory for wind-induced ocean currents which goes beyond the limitations of the classical Ekman spiral. In Rotating spherical coordinates particular, we obtain Ekman-type solutions which extend over large regions in both latitude and longitude; we present examples for constant and for variable eddy viscosities. We also show how an additional restriction on our solution recovers the classical Ekman solution (which is valid only locally). 1. Introduction this approach is still valid only in the neighbourhood of a point on the surface of the sphere, i.e. it is purely local. Of course, the formulation of The Ekman spiral (Ekman, 1905) is one of the foundations of physical oceanic flows in rotating, spherical coordinates is not new; see, for example, oceanography, being the basis for the discussion of ocean circulation in non- Marshall et al. (1997) and Veronis (1973).
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